Question

The equation of refraction at a spherical surface is

$\frac{{\mu }_2}{\nu }-\frac{{\mu }_1}{\mu }=\frac{{\mu }_2-{\mu }_1}{R}$.

Taking $R=\infty$, show that this equation leads to the equation

$\frac{\mathrm{Real}\mathrm{depth}}{\mathrm{Apparent}\mathrm{depth}}=\frac{{\mu }_2}{{\mu }_1}$

for refraction at a plane surface.

Answer 1

Proof:
$$\begin{gathered} \frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R} \\ \mathrm{Now}R=\infty \\ \frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{\infty } \\ \frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=0 \\ \frac{{\mu }_2}{v}=\frac{{\mu }_1}{u} \\ \frac{{\mu }_1}{{\mu }_2}=\frac{u}{v} \\ \mathrm{But}\frac{u}{v}=\frac{\mathrm{Real}\mathrm{depth}/\mathrm{height}}{\mathrm{Apparent}\mathrm{depth}/\mathrm{height}} \\ \therefore \frac{\mathrm{Real}\mathrm{depth}/\mathrm{height}}{\mathrm{Apparent}\mathrm{depth}/\mathrm{height}}=\frac{{\mu }_{\mathit{1}}}{{\mu }_{\mathit{2}}} \end{gathered}$$